Quantification correction for free-breathing myocardial T1ρ mapping in mice using a recursively derived description of a T1ρ* relaxation pathway

Background Fast and accurate T1ρ mapping in myocardium is still a major challenge, particularly in small animal models. The complex sequence design owing to electrocardiogram and respiratory gating leads to quantification errors in in vivo experiments, due to variations of the T1ρ relaxation pathway. In this study, we present an improved quantification method for T1ρ using a newly derived formalism of a T1ρ* relaxation pathway. Methods The new signal equation was derived by solving a recursion problem for spin-lock prepared fast gradient echo readouts. Based on Bloch simulations, we compared quantification errors using the common monoexponential model and our corrected model. The method was validated in phantom experiments and tested in vivo for myocardial T1ρ mapping in mice. Here, the impact of the breath dependent spin recovery time Trec on the quantification results was examined in detail. Results Simulations indicate that a correction is necessary, since systematically underestimated values are measured under in vivo conditions. In the phantom study, the mean quantification error could be reduced from − 7.4% to − 0.97%. In vivo, a correlation of uncorrected T1ρ with the respiratory cycle was observed. Using the newly derived correction method, this correlation was significantly reduced from r = 0.708 (p < 0.001) to r = 0.204 and the standard deviation of left ventricular T1ρ values in different animals was reduced by at least 39%. Conclusion The suggested quantification formalism enables fast and precise myocardial T1ρ quantification for small animals during free breathing and can improve the comparability of study results. Our new technique offers a reasonable tool for assessing myocardial diseases, since pathologies that cause a change in heart or breathing rates do not lead to systematic misinterpretations. Besides, the derived signal equation can be used for sequence optimization or for subsequent correction of prior study results. Supplementary Information The online version contains supplementary material available at 10.1186/s12968-022-00864-2.

In the present work, a novel signal equation for T 1ρ quantification was derived. The derivation, which was based on the solution of a recursion problem, can be followed step-by-step in the attached Online Files (Mathematica Notebook). It has been found that the deviation from the simple monoexponential model can be expressed by a single global sequence parameter λ (dimensionless [0. . . 1]). Knowing λ, T 1ρ can be determined with the corrected fit. If the monoexponential model is used instead, systematically smaller values of T * 1ρ are obtained. Online Figure 1 shows the ratio r * ρ = T * 1ρ /T 1ρ as a function of λ from the data of a lookup table. Here it can be seen that small values λ lead to small quantification errors, while large values lead to high systematic deviations. The r * ρ (λ) relationship was numerically described by a polynomial fit. The results of this fit can be used to determine the approximated value T approx 1ρ for a measured T * 1ρ at an arbitrary λ.

Online Equation 1
Subsequent correction of study results The approximated T 1ρ correction was exemplarily applied to myocardial T 1ρ mapping in mice. Here, the results of T * 1ρ and T 1ρ (already presented in Figure 7) were supplemented with subsequently corrected T approx 1ρ values (Online Figure 2). For each data point, the T * 1ρ value (monoexponential fit) was used to calculate an approximated T approx 1ρ value based on the corresponding λ and Online Equation 1. Here, λ was calculated using the averaged T rec during the complete data acquisition. Thus, the corrected fitting approach is not necessary and the approximated T approx 1ρ values can be subsequently calculated from the underestimated T * 1ρ values. This technique yields values that show good agreement with the corrected T 1ρ values. However, the standard deviation of T approx 1ρ (±2.8%) is slightly higher than the standard deviation of the corrected values (±2.3%). Online Figure 3 examines whether there is a correlation between the discrepancy (T approx 1ρ vs T 1ρ ) and respiration. It can be seen that there is no correlation with T rec itself (Online Figure 3A, r = 0.068, not significant). However, we found a positive correlation with the drift of recovery time during data acquisition (Online Figure 3B, r = 0.865, p < 0.001). The drift values were in a range of -2. . . 1ms/cycle for the N=44 measurements. The discrepancy (T approx 1ρ vs T 1ρ ) was between -5. . . 3%. If there is no drift, T approx 1ρ ≈ T 1ρ . This result indicates that approximated T 1ρ calculation is in principle an improvement over T * 1ρ . However, the results can be systematically influenced by a drift of the respiration during the measurement. Thus, a fit with the new signal equation (Equation 5) considering the respiratory drift (Equation 11) is preferable, provided that the physiological data were recorded completely. For an approximated correction (based on an averaged T rec ), we have provided a computable interactive document (Online File: .cdf).

Online Figures
Online Figure 1: The ratio r * ρ = T * 1ρ /T 1ρ was calculated as a function of the global sequence parameter λ. A lookup table was calculated based on 100 different values of λ and for 100 different T 1ρ values in the range 10...200ms. No direct dependence of the ratio on T 1ρ itself could be found. The data of the lookup table were approximated with a 5th order polynomial fit (R 2 > 0.999). The results of this approximation were used for subsequent correction of study results (Online Figure 2).
Online Figure 2: Comparison of myocardial T 1ρ mapping in mice based on monoexponential fitting, corrected fitting and the subsequent correction using the approximated ratio r * ρ (λ). The approximated values clearly separate from T * 1ρ and show good agreement with the corrected T 1ρ values fitted by the novel signal equation 5. However, a slightly higher variation is observed (±2.8% vs ±2.3%) for the approximation. The correlation with T rec is not significant (r=0.068).
Online Figure 3: Correlation of the discrepancy (T approx 1ρ vs T 1ρ ) with the recovery time T rec (A) and the drift of recovery time (B). There is no correlation with T rec itself. However, with the drift a clear positive correlation can be proven. The drift values were determined from the slope of the linear regression of the T rec courses. The units of the drift values (ms/cycle) indicates how large the change in T rec is per respiratory cycle. In each respiratory cycle, one spin-lock preparation was performed followed by NR=4 readouts in the diastole. Since first short and then long spin-lock times were sampled and the recovery time also depends on the spin-lock time (Equation 11), the drift values tend to negative values.